By the last decade of the twentieth century the Boltzmann flow model enters in maturity, starting by the work of Desvilletes [1]} who proved interesting point that it can propagate any moments, and shortly after, Wennberg [2]} showed that moments can be generated if the initially data has bounded entropy in addition to bounded mass and energy. In the same year of Wennberg's work, a groundbreaking result of Bobylev [3]} showed that the generated moments are summable paving the way to the understanding of propagation and generation of \(L^1({\mathbb {R}}^d)\)  exponentially weighted estimates in several frameworks such us inelastic theories for hard spheres by
Gamba Panferov and Bobylev, Alonso and Lods for hard spheres in [4]}, [5]}, [6]} and later with Panferov, Gamba and Villani [7]} showed the extension of this estimates to hard potentials and show the technique extends to angular transition functions in \(L^1({\mathcal {S}}^d)\)  as well establishing that \(L^\infty ({\mathbb {R}}^d)\) -Maxwellian weighted estimates for solutions as constructed by Arkeryd [8]}, later extended by Bobylev and Gamba [9]} to the Maxwell type of interactions.
These results inspired the work of Mouhot [10]} on generation of \(L^1({\mathbb {R}}^d)\)  exponentially weighted estimates, Alonso and Gamba [11]} solutions derivatives, and Canizo and Mouhot, in collaboration with the authors, [12]}, to study their summability by showing the convergence of moments's partial sums.
